2017 mathematical methods-nht written examination 15 2017 mathmeth exam 1 (nht) turn over question 3...
TRANSCRIPT
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MATHEMATICAL METHODSWritten examination 1
Tuesday 6 June 2017 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.15 pm (1 hour)
QUESTION AND ANSWER BOOK
Structure of bookNumber of questions
Number of questions to be answered
Number of marks
8 8 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof13pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Youmaykeeptheformulasheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
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2017MATHMETHEXAM1(NHT) 2
THIS PAGE IS BLANK
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3 2017MATHMETHEXAM1(NHT)
TURN OVER
Question 1 (4marks)
a. Let y e xx=
2
2cos .
Find dydx
. 2marks
b. Let f :(0,π)→R,where f (x)=loge(sin(x)).
Evaluate ′
f
π3
. 2marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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2017MATHMETHEXAM1(NHT) 4
Question 2 (5marks)a. Findanantiderivativeofcos(1–x)withrespecttox. 1mark
b. Evaluate 3 42 21
2x
xdx+
∫ . 2marks
c. Find f (x)giventhat f(4)=25and ′ = − + >−f x x x x( ) ,38
10 1 0212 . 2marks
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5 2017MATHMETHEXAM1(NHT)
TURN OVER
Question 3 (3marks)
a. Statethesmallestpositivevalueofksuchthat x = 34π isasolutionoftan(x)=cos(kx). 1mark
b. Solve2sin2(x)+3sin(x)–2=0,where0≤x≤2π. 2marks
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2017MATHMETHEXAM1(NHT) 6
Question 4 –continued
Question 4 (5marks)
Let f : ,−
π π2 2
→R,where f(x)=tan(2x)+1.
a. Sketchthegraphof f ontheaxesbelow.Labelanyasymptoteswiththeappropriateequation,andlabeltheendpointsandtheaxisinterceptswiththeircoordinates. 4marks
0
2
–2
–4
4
y
x
2π
−4π
−4π
2π
–1
1
–3
3
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7 2017MATHMETHEXAM1(NHT)
TURN OVER
b. Usefeaturesofthegraphinpart a.tofindtheaveragevalueof f between x = −π8and
x = π8. 1mark
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2017MATHMETHEXAM1(NHT) 8
Question 5 (6marks)Recordsofthearrivaltimesoftrainsatabusystationhavebeenkeptforalongperiod.TherandomvariableXrepresentsthenumberofminutesafterthescheduledtimethatatrainarrivesatthisstation,thatis,thelatenessofthetrain.Assumethatthelatenessofonetrainarrivingatthisstationisindependentofthelatenessofanyothertrain.ThedistributionofXisgiveninthetablebelow.
x –1 0 1 2
Pr(X=x) 0.1 0.4 0.3 p
a. Findthevalueofp. 1mark
b. FindE(X ). 1mark
c. Findvar(X ). 2marks
d. Apassengercatchesatrainatthisstationonfiveseparateoccasions.
Whatistheprobabilitythatthetrainarrivesbefore thescheduledtimeonexactlyfouroftheseoccasions? 2marks
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9 2017MATHMETHEXAM1(NHT)
TURN OVER
Question 6 (3marks)Atalargesportingarenathereareanumberoffoodoutlets,includingacafe.
a. Thecafeemploysfivemenandfourwomen.Fourofthesepeoplearerosteredatrandomtoworkeachday.LetP̂representthesampleproportionofmenrosteredtoworkonaparticularday.
i. ListthepossiblevaluesthatP̂cantake. 1mark
ii. FindPr(P̂=0). 1mark
b. Thereareover80000spectatorsatasportingmatchatthearena.FiveinnineofthesespectatorssupporttheGoannasteam.Asimplerandomsampleof2000spectatorsisselected.
WhatisthestandarddeviationofthedistributionofP̂,thesampleproportionofspectatorswhosupporttheGoannasteam? 1mark
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2017MATHMETHEXAM1(NHT) 10
Question 7 (6marks)Let f :R →R,where f (x)=2x3+1,andletg:R→R,whereg (x)=4–2x.
a. i. Findg (f (x)). 1mark
ii. Find f (g (x))andexpressitintheformk–m(x–d )3,wherem,kanddareintegers. 2marks
b. ThetransformationT:R2 →R2withruleTxy a
xy
bc
=
+
1 00
, wherea,bandcare
integers,mapsthegraphof y=g (f (x))ontothegraphof y=f (g (x)).
Findthevaluesofa,bandc. 3marks
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11 2017MATHMETHEXAM1(NHT)
TURN OVER
CONTINUES OVER PAGE
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2017MATHMETHEXAM1(NHT) 12
Question 8 –continued
Question 8 (8marks)Theruleforafunction f isgivenby f x x( ) = + −2 3 1,where f isdefinedonitsmaximaldomain.
a. Findthedomainandruleoftheinversefunction f –1. 2marks
b. Solve f (x)=f–1(x). 2marks
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13 2017MATHMETHEXAM1(NHT)
END OF QUESTION AND ANSWER BOOK
c. Let g D R g x x c: , ( ) ,→ = + −2 1 whereDisthemaximaldomainofgandcisarealnumber.
i. Forwhatvalue(s)ofcdoesg (x)=g–1(x)havenorealsolutions? 2marks
ii. Forwhatvalue(s)ofcdoesg (x)=g–1(x)haveexactlyonerealsolution? 2marks
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MATHEMATICAL METHODS
Written examination 1
FORMULA SHEET
Instructions
This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
Victorian Certificate of Education 2017
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2017
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MATHMETH EXAM 2
Mathematical Methods formulas
Mensuration
area of a trapezium 12a b h+( ) volume of a pyramid 1
3Ah
curved surface area of a cylinder 2π rh volume of a sphere
43
3π r
volume of a cylinder π r 2h area of a triangle12bc Asin ( )
volume of a cone13
2π r h
Calculus
ddx
x nxn n( ) = −1 x dx n x c nn n=
++ ≠ −+∫ 1 1 1
1 ,
ddx
ax b an ax bn n( )+( ) = +( ) −1 ( ) ( ) ( ) ,ax b dx a n ax b c nn n+ =
++ + ≠ −+∫ 1 1 1
1
ddxe aeax ax( ) = e dx a e cax ax= +∫ 1
ddx
x xelog ( )( ) =1 1 0x dx x c xe= + >∫ log ( ) ,
ddx
ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dx a ax c= − +∫ 1
ddx
ax a axcos( )( ) −= sin ( ) cos( ) sin ( )ax dx a ax c= +∫ 1
ddx
ax aax
a axtan ( )( )
( ) ==cos
sec ( )22
product ruleddxuv u dv
dxv dudx
( ) = + quotient rule ddx
uv
v dudx
u dvdx
v
=
−
2
chain ruledydx
dydududx
=
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3 MATHMETH EXAM
END OF FORMULA SHEET
Probability
Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)
Pr(A|B) = Pr
PrA BB∩( )( )
mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2
Probability distribution Mean Variance
discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)
continuous Pr( ) ( )a X b f x dxa
b< < = ∫ µ = −∞
∞
∫ x f x dx( ) σ µ2 2= −−∞∞
∫ ( ) ( )x f x dx
Sample proportions
P Xn
=̂ mean E(P̂ ) = p
standard deviation
sd P p pn
(ˆ ) ( )= −1approximate confidence interval
,p zp p
np z
p pn
−−( )
+−( )
1 1ˆ ˆ ˆˆˆ ˆ
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3 MATHMETH EXAM
END OF FORMULA SHEET
Probability
Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)
Pr(A|B) = Pr
PrA BB∩( )( )
mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2
Probability distribution Mean Variance
discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)
continuous Pr( ) ( )a X b f x dxa
b< < = ∫ µ = −∞
∞
∫ x f x dx( ) σ µ2 2= −−∞∞
∫ ( ) ( )x f x dx
Sample proportions
P Xn
=̂ mean E(P̂ ) = p
standard deviation
sd P p pn
(ˆ ) ( )= −1approximate confidence interval
,p zp p
np z
p pn
−−( )
+−( )
1 1ˆ ˆ ˆˆˆ ˆ
2017 Mathematical Methods 1InstructionsFormula sheet